Dynamic event-triggered delay compensation control for networked predictive control systems with random delay

This paper aims to investigate the dynamic event-triggered control problem for networked predictive control systems with random delays and disturbance. First, a discrete-time dynamic event-triggered control scheme, in which sensor information is only updated when it is necessary, is presented. Next, the systems are modeled as a time-delay singular Markovian jump systems with time-varying switching. Then, a dynamic event-triggered delay compensation control strategy is proposed. Sufficient conditions guaranteeing the asymptotically stable are derived based on the Lyapunov–Krasovskii functional method together with the linear matrix inequality (LMI) technique. Finally, simulation results verify the effectiveness of the proposed strategy.

• By integrating the network delay compensation approach and ETC, a dynamic event-triggered network delay compensation control strategy is proposed.The method combines the merits of reducing the networked bandwidth occupation and compensating for network delay actively.• Based on the predictive control, delay-dependent state feedback controllers with time-varying gains are designed.Which have less conservatism than the fixed gain predictive controller 29 .• The stability and stabilization problems of the dynamic event-triggered NCSs with and without external disturbance are discussed.The H ∞ performance of the system under biased noise is guaranteed.
The rest of the article is organized as follows."Problem formulation" states the problem formulations."Main results" gives the stability analysis and controller design."Robust control of NPC" shows the H ∞ performance based on the proposed method.Simulations are shown in "Simulation and experiment results".A discussion is shown in "Discussion", Finally, "Conclusion" presents a conclusion to the paper.

Problem formulation
Considering the following discrete-time system: where x(k) ∈ R n , y(k) ∈ R q , and u(k) ∈ R m are the state vector, output vector and input vector, respectively.ω(k) ∈ R m is the disturbance input.A, B, C, D and E are the constant matrices of appropriate dimensions.The structure of the NCSs is shown in Fig. 1.On the sensor side, the DETM is added to the system.It is assumed that the state data x(k s ) is transmitted successfully at the time k s , the following condition is introduced 28 where r = {1, 2, . . ., N} .0 < µ < 1 is a given scalar parameter.and θ(k) is determined by (1) where β > 1 , θ(0) > 0 .Supposing that the first state x(0) is transmitted successfully.When the above condition is satisfied, then the state is transmitted, otherwise, it cannot be sent.The system executes the last data and then generates a new state that is verified whether it satisfies the condition (2).
The sequence of states which satisfy the above condition are packed and sent to the controller With the traditional state feedback controller u(k s ) = Kx(k s ) , where the gain K is fixed for all the time.In consideration of the time variability of networked-induced delay, a more reasonable control law is designed 28 where, the delay τ k = {0, 1, 2 . . ., τ } , the feedback gain K(τ k ) is switched depend on networked-induced delay.Depending on the different delays, there exists a sequence control signal as follows: where the τ is the upper bound of the time-delay.We can select the suitable control signal based on the delay τ k at any time k s .The control signals were packed and transmitted to the actuator side in the form as On the actuator side, the NDC chooses the suitable control signal depending on the networked-induced delay.The relationship between the delay τ k , the current time k and the event-triggered time k s is And the predictive controller is Then, when the ω(k) = 0 , the closed-loop system is Referring to the article 30 , the relationship between triggering time and networked delay is divided into the following two cases.
Case 1: where ) Then it can be easily shown that In Case 1, define e(k) = 0 .In Case 2,define Combine the e(k) and the event-triggered condition (2), we have After that, for k ∈ [k s + τ k s , k s+1 + τ k s+1 ) , closed-loop system (1) without disturbance can be further rewritten as The purpose of this paper is to propose appropriate control strategies to make the above systems still operate stably under the influence of delay and interference, and the frequency of data transmission can be reduced.
Remark 1 As we all know, Zeno behavior refers to an event triggering an infinite number of times for a finite period of time, which can occur in the study of event triggering in continuous-time systems.But in a discretetime control system, the worst-case scenario for event triggering is that data is sent at every moment k, i.e. time triggering.Therefore, even if the event trigger of the discrete-time system occurs in the Zeno behavior, the worst case will disable its ability to reduce the frequency of data transmission, but it will not affect the stability of the system.

Main results
In this section, the stability and controller design are addressed.Selecting a switched Lyapunov-Krasovskii function is to to prove the stability of the system.

Stability analysis
Firstly, the stability problem of closed-loop NPC systems without disturbance is addressed.
Theorem 1 For given parameter µ > 0 and gains K i , system (17) is asymptotically stable if there exist real matrices Proof Construct the following Lyapunov-Krasovskii function as where δ(l) = x(l + 1) − x(l).
From Eq. ( 16), for all k ∈ (k s , k s+1 ), Then, it can be obtained that and θ(0) > 0 , we have θ(k) > 0 Along the solution of system (17), we obtain that (18) www.nature.com/scientificreports/Based on the free-weighting matrix approach 31 , it can be seen that Combing the Eqs.( 25), ( 26), ( 27), ( 28), we have where Further, let P j ≤ P i , Q j ≤ Q i , R j ≤ R i , the formula (29) can be simplified to where The condition of theorem one can be made by Shure's complement theorem.If condition (18-20) is met, then �V i (k) < 0 , i.e. system ( 17) is asymptotically stable.The proof is completed.

Remark 2
The Lyapunov-Krasovskii function introduces a time-delay term, which makes it less conservative in analyzing the stability of time-delay systems.The value of the Barrier Lyapunov function (BLF) tends to infinity as the function variable approaches the constraint boundary, as shown in the figure.This means that while the BLF remains bounded by the designed controller, the function variables remain within the constraint boundary, i.e. the constraint is satisfied.This makes BLF more advantageous in dealing with systems with state constraints.

Controller design
Based on Theorem 1, the delay-dependent controller design method for the system (17) can be easily obtained.
Theorem 2 For given parameters > 1 , µ > 0 and d m , under the event-triggering condition (2, 3), the system in (17) is stabilizable, if there exist matrices Qi > 0 , W i > 0 , V i > 0 , Si > 0 , �i > 0 , Xi , Ȳi , Ki with appropriate dimensions satisfying the following LMIs: In addition, by solving LMI and matrix transforma- tion, the switched controller gains matrices are given by K i = Ki W −1 i .
Proof Pre-and post-multiplying Eqs. ( 18) , (19) with , respectively.and defining some new variables as The Eq. ( 30) can be obtained.The proof is completed.

Robust control of NPC
Now, we are in a position to solve the problems of stability of NPC system with disturbance and H ∞ controller design of NPC system with event-triggered mechanism.The networked closed-loop system with disturbance is obtained that

Controller design
Similar to the Theorem 2, the switched H ∞ controller gains K i for system (28) are calculated by the following theorem.
Theorem 4 For given parameters > 1 , µ > 0γ and d m , under the event-triggering condition (2, 3), the system in Eq. ( 34) is stabilizable, if there exist matrices Qi > 0 , W i > 0 , V i > 0 , Si > 0 , �i > 0 , Xi , Ȳi , Ki with appropriate dimensions satisfying the following LMIs: In addition, by solving LMI and matrix transformation, the switched controller gains matrices are given by K The proof is similar to the Theorem 2, thus omitted.

Simulation and experiment results
In this section, two numerical examples are given to verify the effectiveness of the approach we presented.

Example 1: stabilization of the networked system without disturbance
In this case, a discrete-time linear system without disturbance is considered as the following The parameters mentioned in Theorem 2 are given: To solve the Theorem 2 by using LMI, the time-varying gains and the weight matrix can be obtained as follows www.nature.com/scientificreports/Considering the 0-20 steps random RTT delay shown in Fig. 2, a comparison of the results of the two eventtriggered control methods (SETC and DETC) is shown in Fig. 3. Obviously, with similar stability results, DETC has fewer trigger moments than SETC.
In addition, the comparison of the methods between NPC with fixed gains and time-varying gains is given.The simulation results is shown in Fig. 4. As can be seen from the pictures, even if the system has random delay, the method we proposed can still make the system eventually stable and reduce the frequency of data transmission.In addition, we compare the proposed method with the fixed-gain NPC method, and the results show that our method can converge faster.

Example 2: stabilization of the networked event-triggered control system with disturbance
In this case, the robust H ∞ control of NCSs: the ETC and NDC approach is verified, the system parameters is following The disturbance is set to be ω(k) = sign(sin(k)), k < 30 0, k ≥ 30 The random delay is shown in the Fig. 5.And the parameters in event-triggered condition (3) and the Theorem 4 are d m = 12, = 2, µ = 0.5, γ = 0.01, θ(0) = 0.01, β = 5 .Then the controller gains and the parameter can be solved by MATLAB LMI toolbox   In the case of disturbance and delay, the system state and event trigger interval are shown in Fig. 6.As can be seen from Figs. 6a-d, the system eventually remains stable despite some jitter under disturbance and delay.And the number of data transfers is reduced, and the DETC method performs better than SETC.The simulation results in this part show that the proposed method can still make the system converge in the presence of delay and disturbance.

Discussion
This paper combines dynamic event trigger control and networked predictive control methods and presents a delay compensation control scheme based on dynamic event triggering.This solution allows the system to operate stably under the influence of delay and disturbance and reduces data transmission.Compared with the static event triggering method, the method proposed in this article can further reduce data transmission without affecting system stability.The introduction of event gain avoids the control of fixed gain for better performance.The scheme based on networked predictive control can actively compensate for the delay, which is less conservative than the traditional method.
LMI can only get sufficient conditions to stabilize the system.Secondly, LMI is an optimization tool, but multiple matrices are scaled in the stability analysis, so the result may be a suboptimal solution.In addition, delay-dependent controller gain switching improves control performance, but it is difficult to derive its globally stable conditions.Issues such as less conservative adequacy and overall optimal performance will be the focus of our future research.

Conclusion
This paper has addressed the DETC problem of discrete-time networked predictive control systems with simultaneous consideration of delays and disturbance.The closed-loop system is obtained by investigating the NDC method, the DETC method, and the feedback time delay-dependent gain method.The DETC method can effectively reduce the network bandwidth resources occupied by data transmission.The random delay has compensated by NDC actively.A dynamic event-triggered network delay compensation control strategy has proposed.Then, the delay-dependent stability conditions have been derived by using the LMI approach.Based on these conditions, the time-varying gain predictive controller has designed.Furthermore, the robust H ∞ control prob- lem of NCSs has discussed.Finally, simulations illustrate the effectiveness of the proposed algorithm.

Figure 4 .
Figure 4.The results of x 1 between NPC and our method.